http://cseweb.ucsd.edu/~dasgupta/103/2b.pdf part 2.1.2 implies $P(X|Y \cap Z) = \frac{P(X|Y)}{P(Y|Z)}$
Seems to imply that this is true but if you take bayes, the left hand side is: $P(X|Y \cap Z) = \frac{P(X \cap Y \cap Z)}{P ( Y \cap Z)}$
While the right hand side is: $\frac{P(X|Y)}{P(Y|Z)} = \frac{\frac{P(X \cap Z)}{P(Z)}}{\frac{P(Y \cap Z)}{P(Z)}} = \frac{P(X \cap Z)}{P(Y \cap Z)}$
This seems to imply that $P(X \cap Z) = P(X \cap Y \cap Z)$ but this isnt always true??
As you observed, it is not necessarily the case that $\Pr(X|(Y\cap Z))=\frac{\Pr(X|Y)}{\Pr(Y|Z)}$.
For example, toss a fair coin twice. Let $X$ and $Y$ each be the events "first is a head," and let $Z$ be the event "second is a head." Then $\frac{\Pr(X|Y)}{\Pr(Y|Z)}=2$, so it is not even a probability.